materials

Review

Coherence in the Ferroelectric A3ClO (A = Li, Na) Familyof Electrolytes

Maria Helena Braga

Citation: Braga, M.H. Coherence in

the Ferroelectric A3ClO (A = Li, Na)

Family of Electrolytes. Materials 2021,

14, 2398. https://doi.org/10.3390/

ma14092398

Academic Editor: Yaniv Gelbstein

Received: 14 March 2021

Accepted: 29 April 2021

Published: 5 May 2021

Publisher’s Note: MDPI stays neutral

with regard to jurisdictional claims in

published maps and institutional affil-

iations.

Copyright: © 2021 by the author.

Licensee MDPI, Basel, Switzerland.

This article is an open access article

distributed under the terms and

conditions of the Creative Commons

Attribution (CC BY) license (https://

creativecommons.org/licenses/by/

4.0/).

LAETA-INEGI, Engineering Physics Department, Engineering Faculty, University of Porto,R. Dr. Roberto Frias s/n, 4200-465 Porto, Portugal; mbraga@fe.up.pt

Abstract: Coherence is a major caveat in quantum computing. While phonons and electrons areweakly coupled in a glass, topological insulators strongly depend on the electron-phonon coupling.Knowledge of the electron−phonon interaction at conducting surfaces is relevant from a funda-mental point of view as well as for various applications, such as two-dimensional and quasi-1Dsuperconductivity in nanotechnology. Similarly, the electron−phonon interaction plays a relevantrole in other transport properties e.g., thermoelectricity, low-dimensional systems as layered Bi andSb chalcogenides, and quasi-crystalline materials. Glass-electrolyte ferroelectric energy storage cellsexhibit self-charge and self-cycling related to topological superconductivity and electron-phononcoupling; phonon coherence is therefore important. By recurring to ab initio molecular dynamics,it was demonstrated the tendency of the Li3ClO, Li2.92Ba0.04ClO, Na3ClO, and Na2.92Ba0.04ClOferroelectric-electrolytes to keep phonon oscillation coherence for a short lapse of time in ps. Double-well energy potentials were obtained while the electrolyte systems were thermostatted in a heatbath at a constant temperature. The latter occurrences indicate ferroelectric type behavior but donot justify the coherent self-oscillations observed in all types of cells containing these families ofelectrolytes and, therefore, an emergent type phenomenon where the full cell works as a feedbacksystem allowing oscillations coherence must be realized. A comparison with amorphous SiO2 wasperformed and the specific heats for the various species were calculated.

Keywords: decoherence; ferroelectrics; energy storage; electrolytes; van der Pol oscillators; Duffingoscillators; Poincaré maps; FitzHugh-Nagumo model; Hodgkin-Huxley model; Rabi oscillations

1. Introduction

Statistical physics relates the average behavior of a physical system with its micro-scopic constituents and their interactions. Under certain conditions, the equilibriumprobability of a microscopic configuration x (e.g., all spin configurations, positions of allatoms, molecules, ions, etc.) is proportional to e−E(x), the Boltzmann distribution. Thedimensionless energy E(x) contains the potential energy of the system, the temperature(thermal energy kT, where T\displaystyle T is temperature, and k\displaystyle k is theBoltzmann constant), and other thermodynamic quantities [1].

Quantum mechanics offers an accurate description of a broad variety of physicalsystems. Nonetheless, a demonstration that quantum mechanics applies equally to macro-scopic mechanical systems has been a long-abiding challenge [2].

Quantum Ising models (Figure 1a–c) are canonical models for the study of quan-tum phase transitions [3] and are the basic concept for many analog quantum computingand quantum annealing ideas [4]. Recent experiments with cold atoms have reached theinteraction-dominated regime in quantum Ising magnets via optical coupling of trappedneutral atoms to Rydberg states [5]. Rydberg atoms have several unique properties includ-ing an overemphasized response to electric and magnetic fields. With the application ofan external electric field, Rydberg atoms might develop exceptionally large electric dipolemoments making them very susceptible to perturbation by the field [6]. Strong correlations

Materials 2021, 14, 2398. https://doi.org/10.3390/ma14092398 https://www.mdpi.com/journal/materials

Materials 2021, 14, 2398 2 of 16

in quantum Ising models have been observed in several experiments, starting from a singleexcitation in the superatom regime up to the point of crystallization. Rapid developmentsin this field make spin systems based on Rydberg atoms a promising platform to quantumsimulation because of the unmatched flexibility and strength of interactions combined withhigh control and good isolation from the environment [7].

Spin Hamiltonians are repeatedly introduced as compliant simplifications of realisticcondensed matter Hamiltonians and have been hugely successful in describing manymaterials [4,6–8]. Recently, they have attracted interest because they are a well-suited basefor analog quantum computation (Figure 1c) [9–13].

Decoherence has been utilized to understand the collapse of the wave function inquantum mechanics. Specifically, components of the wave function are decoupled froma coherent system and acquire phases from their immediate surroundings. A total super-position of the global or universal wave function still exists (and remains coherent at theglobal level), but its fate remains an interpretational issue. Decoherence does not attemptto explain the measurement problem. Rather, decoherence explains the transition of thesystem to a mixture of states that seem to correspond to those states observers perceive.Decoherence embodies a challenge for the realization of quantum computers since suchmachines are expected to rely on the undisturbed evolution of quantum coherences. Simplystated, they require that the coherence of states be maintained and that decoherence ismanaged, to perform quantum computation. The preservation of coherence, and mitigationof decoherence effects, are therefore linked to the concept of quantum error correction [13].

Rabi oscillations arise when a few-level quantum system is exposed to a periodicallytime-varying field Figure 1d [14]. These oscillations are pervasive in many areas of physics,from atomic [15,16] and condensed-matter physics to quantum information science, havingbeen reported for Rydberg atoms [17], nuclear spin states of different exchange symme-try [18], cavity polaritons [19–21], impurity states in insulators [22], excitons [23], and spinstates in quantum dots [24,25] as well as Josephson junctions [26]. Floquet states (Figure 1d)are stationary solutions of a quantum system in the subjected to a periodic time-dependentperturbation. Such states have enticed considerable attention recently, arising primarilyfrom the possibility that new phases with interesting topological properties may emergeunder high-intensity excitation [26–32].

If a system’s parameter is gradually varied, it produces gradual variations in the stateof the system itself. There can be circumstances, however, in which gradual variations ofa parameter cause sudden or catastrophic changes of state. A system of this kind showsbifurcations [10] (Figure 1e). Bifurcation theory has been implemented to correlate quantumsystems to the dynamics of their classical analogs in atomic systems [33–38], molecularsystems [39], and resonant tunneling diodes [40]. Bifurcation theory has also been appliedto the study of laser dynamics [41], and many theoretical examples, which are hard to reachexperimentally such as the kicked top [42] and coupled quantum wells [43]. The leadingreason for the link between quantum systems and bifurcations in the classical equationsof motion is that at bifurcations, the mark of classical orbits becomes large [44,45]. Manykinds of bifurcations have been studied with respect to the associations between classicaland quantum dynamics including saddle node bifurcations, Hopf bifurcations, umbilicbifurcations, period doubling bifurcations, reconnection bifurcations, tangent bifurcations,and cusp bifurcations [45].

Materials 2021, 14, 2398 3 of 16Materials 2021, 14, x 3 of 16

(a) (b)

(c) (d) (e)

(f)

(g)

Figure 1. Ferroelectric, ferromagnetic, qubit, Ising model, Rabi oscillation, bifurcation, and damped-forced oscillator. (a) Landau–Ginzburg–Devonshire model for a ferroelectric. Left, the double-well free energy F in a ferroelectric as a function of the electric polarization P. Red shading denotes regions of negative capacitance (C < 0). Right, ferroelectric polarization–electric field / potential difference dependence obtained from the free energy F. The ‘S’-shaped P vs EField curve displays a region of negative capacitance related to the energy barrier of F. Arrows indicate ferroelectric polarization directions; (b) Double-well potential energy diagram and the lowest quantum energy levels corresponding to the qubit. States |↑ > and |↓ > are the lowest two levels, respectively. The intra-well energy spacing ωp. The measurement detects magnetization and does not distinguish between, |↑ > and excited states within the right-hand well. These excitations are exceedingly improbable at the time the state is measured [10]. (c) Minimal superatom with three spins. The configurations are coupled by spin-flips with coupling Ω (grey arrows). The interaction energy of each state is indicated by the brightness of the grey background. When selecting states with a minimal distance between spin ↑, all states in the red shaded area are neglected. While the coupling from the ground state ∣↓↓↓ > to the states with n↑ = 1 is spatially homogeneous, the coupling of the singly excited states to the state with n↑ = 2 is spatially inhomogeneous [6]; (d) Fouquet states; model of a harmonically

Figure 1. Ferroelectric, ferromagnetic, qubit, Ising model, Rabi oscillation, bifurcation, and damped-forced oscillator.(a) Landau–Ginzburg–Devonshire model for a ferroelectric. Left, the double-well free energy F in a ferroelectric as afunction of the electric polarization P. Red shading denotes regions of negative capacitance (C < 0). Right, ferroelectricpolarization–electric field/potential difference dependence obtained from the free energy F. The ‘S’-shaped P vs EField

curve displays a region of negative capacitance related to the energy barrier of F. Arrows indicate ferroelectric polarizationdirections; (b) Double-well potential energy diagram and the lowest quantum energy levels corresponding to the qubit.States |↑ > and |↓ > are the lowest two levels, respectively. The intra-well energy spacingωp. The measurement detectsmagnetization and does not distinguish between, |↑ > and excited states within the right-hand well. These excitations areexceedingly improbable at the time the state is measured [10]. (c) Minimal superatom with three spins. The configurationsare coupled by spin-flips with coupling Ω (grey arrows). The interaction energy of each state is indicated by the brightnessof the grey background. When selecting states with a minimal distance between spin ↑, all states in the red shaded areaare neglected. While the coupling from the ground state |↓↓↓> to the states with n↑ = 1 is spatially homogeneous, thecoupling of the singly excited states to the state with n↑ = 2 is spatially inhomogeneous [6]; (d) Fouquet states; model ofa harmonically time-varying classical field of amplitude |F0| that couples the ground state |g> to the excited state |e>of a two-level system, which itself is coupled to a continuum of states |χE > [14]; (e) Bifurcation diagram for yn [33]; (f)Duffing oscillator; displacement vs time and velocity vs displacement (phase portrait) [34]; (g) Frequency response z/A as afunction of ω/

√α for the Duffing equation, with α = A = 1 and damping δ = 0.1 (Equation (1)). The dashed parts of the

frequency response are unstable [35]. All the graphs cited were adapted from the reference.

Materials 2021, 14, 2398 4 of 16

One of the dynamical systems exhibiting chaos is the non-linear forced oscillator:

..x + δ

.x + αx + βx3 = A cosωt (1)

where x is an extensive variable such as the displacement or the charge, δ is the dampingparameter, α controls the linear stiffness, β controls the amount of non-linearity in therestoring force; if β = 0, the Duffing equation describes a damped and driven simpleharmonic oscillator, and f = A cosωt is the driving term, A is the amplitude of the periodicdriving force and ω the angular frequency. If β 6= 0, the oscillator is a forced-dampedDuffing oscillator (e.g., Figure 1f,g) [46,47].

Nonlinearity does not always act as a vehicle of disorder or chaos. In fact, it canhave exactly the opposite effect; an example of that, are van der Pol’s self-oscillations withlimit cycles.

The relaxation oscillations have become the foundation of geometric singular perturba-tion theory and play a significant role in the analysis presented here. The circuit showingrelaxation oscillations consists of a feedback loop containing a switching device such asa transistor, comparator, relay [48], or a negative resistance device like a tunnel diode,that repetitively charges a capacitor or an inductor through a resistance until it reaches athreshold level and then discharges again [49,50]. The period of the oscillator depends onthe time constant of the capacitor and/or inductor circuit [51]. The active device switchesabruptly between charging and discharging modes and thus produces a discontinuouslychanging repetitive waveform [49,51,52].

Van der Pol proposed a version of a relaxation oscillations circuit that includes aperiodic forcing term (Figure 2):

..x + ε

(x2 − 1

) .x + x = A sin ωt (2)

Equation (2) can be written as: .x = y

.y = −x + ε

(1− x2)y + A cos(ωt + ϕ)

(3)

In equations such as Equations (2) and (3), van der Pol and van der Mark [53]noted the existence of two stable periodic solutions with different periods for a particularvalue of the parameters and observed noisy behavior in an electrical circuit describedby Equation (2) [53]. In their investigations of the oscillator behavior in the relaxationoscillations regime, they found that the subharmonic oscillations could be observed duringchanges in the natural frequency of the system [52]. Moreover, the authors noted “irregularnoise” before the transition from one subharmonic regime to another. Their paper is per-haps one of the first experimental reports of chaos-which they did not engage in in moredetail [52,53].

Since its introduction in the 1920s, the van der Pol equation has been studied forsystems with self-excited limit cycle oscillations. The classical experimental setup of thesystem is the oscillator with a vacuum triode. The equation has been studied over wideparameter regimes, from perturbations of harmonic motion to relaxation oscillations. TheVan der Pol equation is now used as a basic model for oscillatory processes in physics,electronics, biology, neurology, sociology, and economics [54–59]. Van der Pol himself builtsome electronic circuit models of the human heart to study the range of stability of heartdynamics. His investigations with adding an external driving signal were like the situationin which a real heart is driven by a pacemaker. Van der Pol was interested in finding outhow to stabilize a heart’s irregular beating or “arrhythmias”. Since then, his equivalentcircuit has been used by scientists to model a variety of physical and biological phenomena.For instance, in biology, the van der Pol equation has been used as the basis of a model ofa couple of neurons in the gastric mill circuit of the stomatogastric ganglion [60,61]. TheHodgkin-Huxley and FitzHugh-Nagumo equation is a planar vector field that extends

Materials 2021, 14, 2398 5 of 16

the van der Pol equation as a model for action potentials of neurons (Figure 2f,g). Inseismology, the van der Pol equation has been utilized in the development of the model ofthe interaction of two plates in a geological fault [62].

Materials 2021, 14, x 5 of 16

phenomena. For instance, in biology, the van der Pol equation has been used as the basis of a model of a couple of neurons in the gastric mill circuit of the stomatogastric ganglion [60,61]. The Hodgkin-Huxley and FitzHugh-Nagumo equation is a planar vector field that extends the van der Pol equation as a model for action potentials of neurons (Figure 2(f,g)). In seismology, the van der Pol equation has been utilized in the development of the model of the interaction of two plates in a geological fault [62].

(a)

(b)

(c)

(d) (e)

(f) (g)

Figure 2 Van der Pol oscillator; (a) electrical circuit of a van der Pol oscillator with a tunnel diode showing negative resistance; (b) I vs V characteristic 𝐼 = 𝑓(𝑉) of the tunnel diode (curve 1) and load line 𝐼 = (𝐸 – 𝑉)/𝑅 for a constant current (line 2). These two lines intersect at the operation point 𝑉 = 𝑉 , 𝐼 = 𝐼 ; c) equivalent circuit of the tunnel-diode oscillator for alternating current with a negative resistor –R- (adapted from [63]); (d) the x vs time series of Equations (2) and (3), Poincaré map, and phase portrait (projec-tion of the extended 3D portrait) for values A = −50, 𝜀 = 5 and 𝜔 = 7, corresponding to almost periodic orbit (obtained in this study using wxMaxima 20.06.6 and the Runge-Kutta method, supplementary information) - Poincaré map adapted from [52]; (e) partition-ing of the state space and the Poincaré maps [48]; (f) noisy Hodgkin-Huxley model; time evolution of the three ion channel variables n, m, and h [64]; (g) time evolution of the membrane potential and the adaptation variable in the noisy FitzHugh-Nagumo model (V – voltage, w – current).(f,g) adapted from [64];

2. Contextualizing to the Ferroelectric Glass-Electrolyte Cells In recent years, we have been observing different manifestations of the van der Pol-

Duffing Hodgkin-Huxley and FitzHugh-Nagumo oscillators, in different architectures of electrochemical/electrostatic energy harvesting and storage cells [65–71]. We have studied

Figure 2. Van der Pol oscillator; (a) electrical circuit of a van der Pol oscillator with a tunnel diode showing negativeresistance; (b) I vs V characteristic I = f (V) of the tunnel diode (curve 1) and load line I = (E0–V)/R for a constant current(line 2). These two lines intersect at the operation point V = V0, I = I0; (c) equivalent circuit of the tunnel-diode oscillator foralternating current with a negative resistor –R-(adapted from [63]); (d) the x vs time series of Equations (2) and (3), Poincarémap, and phase portrait (projection of the extended 3D portrait) for values A =−50, ε = 5 and ω = 7, corresponding to almostperiodic orbit (obtained in this study using wxMaxima 20.06.6 and the Runge-Kutta method, supplementary information) -Poincaré map adapted from [52]; (e) partitioning of the state space and the Poincaré maps [48]; (f) noisy Hodgkin-Huxleymodel; time evolution of the three ion channel variables n, m, and h [64]; (g) time evolution of the membrane potential andthe adaptation variable in the noisy FitzHugh-Nagumo model (V—voltage, w—current). (f,g) adapted from [64].

2. Contextualizing to the Ferroelectric Glass-Electrolyte Cells

In recent years, we have been observing different manifestations of the van der Pol-Duffing Hodgkin-Huxley and FitzHugh-Nagumo oscillators, in different architectures ofelectrochemical/electrostatic energy harvesting and storage cells [65–71]. We have studiedwhat seems to be analogous to Rabi’s oscillations arising from Floquet states, namely,

Materials 2021, 14, 2398 6 of 16

stationary solutions of a quantum system in the presence of a periodic time-dependentperturbation [70].

The self-charge, leading to an increase in potential difference and current [70–74],while imparting energy to an electrical load resulting in an mA discharge current, is novelin the energy storage world.

The only way the voltage of an electrochemical cell can increase is if the chemical po-tential µ of the negative electrode increases, the positive decreases, or both (µvaccum = 0 eV).A cell’s voltage upsurge is only accomplished if electrons are conducted from the pos-itive electrode to the negative electrode, which is the charging direction. As the cell isdischarging, the charging current is internal to the cell completing the feedback, as inthe Poincaré maps of Figure 2d,e. Since the electrolyte is an insulator and the electricaldouble-layer capacitors at the interfaces are maintained, the current seems to be topologicaland the alkali mobile ions in the bulk electrolyte do not cease to move towards the positiveelectrode while discharging. Surface electrons tunnel from the electrolyte to the negativeelectrode completing the Poincaré maps. At the negative electrode, Rabi oscillations likethose shown in Figure 1d take place.

We concluded that a topologic supercurrent can be feedback in a ferroelectric-electrolytecell; self-charging materializes by the tunneling of electrons from the electrolytes’ surface tothe negative electrode while the cell is discharging with an external load (such as materialresistors or LEDs) [70–74]. The negative electrode/ferroelectric electrolyte/positive elec-trode cells are then able to maintain self-sustained oscillations above a certain voltage fordays, like those described by the van der Pol and Fitz Hugh-Nagumo models [70,71]. Phe-nomena observed in these ferroelectric-electrolyte cells are similar to those in Figure 2f,gand the physical feedback occurring in the cells can be described as the Poincaré maps ofFigure 2e; a trajectory that has at least a finite number of switchings. Let ξk be the pointon the trajectory where it switches to M+ i.e., ξk ∈ S+. Suppose that after time tk, thetrajectory hits S− at ηk and switches to M−. Let τk, be the time taken for the trajectory tohit S+ again at ξk+1. Suppose that the trajectory is transversal to the switching surfacesat ξk and ηk. Two Poincaré maps ηk = P+(ξk) and ξk+1 = P−(ηk) are then defined. Thecomposition P = P− P+, which maps S+ to itself, defines a Poincaré return map for thefeedback system. Similarly P = P+ P−, which maps S− to itself, is also a Poincaré returnmap (Figure 2e). The fixed points and periodic points of a return map correspond to theperiodic orbits of the feedback system [48].

In the material ferroelectric-electrolyte cells, the S− Poincaré section is the negativeelectrode and the S+ is the positive electrode. ξk is the point on the trajectory where anelectron tunnels from the positive electrode to the surface of the ferroelectric-electrolyte M+

where it is freely conducted (as in a topological superconductor) reaching S− after tunnelingfrom the surface of the electrolyte at ηk, after tk. The trajectory of the electron then switchesto M− which is the external circuit during discharge; for example, a circuit constituted by aload resistor which is connected to the ferroelectric-electrolyte cell where the electron isconducted back to S+ at ξk+1. Then the electron tunnels back to M+ where it is conductedthrough the surface of the ferroelectric to tunnel back to ηk+1 at S− (the negative electrode)a Poincaré return map for the feedback cell corresponding to the experimental phenomenaoccurring in Figures 3 and 4 [70,71].

In our previous studies, it was demonstrated that coherence might be observedwhile discharging electrostatic/electrochemical cells with a load [65–74]. In other words,the cells of the type conductor1/ferroelectric-electrolyte/conductor2 with collective elec-trical self-charge and self-cycling phenomena look like a single macroscopic quantumobject while discharging with a load, as observed in Figures 3 and 4. The ferroelectric-electrolytes (Na2.99Ba0.005ClO and Li2.99Ba0.005ClO) and their polymer composites havehuge dielectric constants such as those observed in Figure 4e–h; the conductor-electrodesin conductor1/ferroelectric-electrolyte/conductor2 do not contain sodium or lithium, inany form, when the cells are assembled. Experiments show that even when discharge is set

Materials 2021, 14, 2398 7 of 16

at relatively high currents such as 0.4–0.2 mA cm−2, coherent self-charge and self-cyclingcan be observed, especially above 35 C [70–74].

Materials 2021, 14, x 7 of 16

(a) (b)

Figure 3. conductor1/ferroelectric-electrolyte/conductor2 cells showing self-charging and self-cycling while the cell is set to discharge. (a) Zn/ferroelectric-electrolyte (Na2.99Ba0.005ClO)/C + Cu cell showing bifurcations corresponding to self-cy-cling while discharging with a resistor of 1000 Ω at 60 °C as in Figure 1e (adapted from [70]); (b) Cu/ferroelectric-electrolyte (Na2.99Ba0.005ClO)/Cu cell after quick charge followed by discharge at open-circuit voltage to 1.1–1.3 V, for then self-charg-ing to 2.3–2.4 V after 6 or 8 h up to approximately 9 or 5 V, respectively (adapted from [65]).

In our previous studies, it was demonstrated that coherence might be observed while discharging electrostatic/electrochemical cells with a load [65–74]. In other words, the cells of the type conductor1/ferroelectric-electrolyte/conductor2 with collective electrical self-charge and self-cycling phenomena look like a single macroscopic quantum object while discharging with a load, as observed in Figures 3 and 4. The ferroelectric-electrolytes (Na2.99Ba0.005ClO and Li2.99Ba0.005ClO) and their polymer composites have huge dielectric constants such as those observed in Figure 4e–h; the conductor-electrodes in conduc-tor1/ferroelectric-electrolyte/conductor2 do not contain sodium or lithium, in any form, when the cells are assembled. Experiments show that even when discharge is set at rela-tively high currents such as 0.4–0.2 mA cm-2, coherent self-charge and self-cycling can be observed, especially above 35 °C [70–74].

(a) (b)

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

1.2

t (hrs)

Pote

ntia

l (V)

0 2 4 6 8 10 120123456789

1011

burst discharge

10 min. charge at 1.2 mA 1 h charge at 0.50 mA 2 h charge at 0.15 mA

burst discharge

t (hrs)

Po

tent

ial (

V) burst discharge

Discharge at OCV

2.4 V 2.3 V

0

1

2

3

4

5Vmea

Icon

Pote

ntia

l (V)

t (hrs)

I (m

A)

Imea

2.4 V

0 10 20 30 40 50 60 70-0.00200.0020.0040.0060.0080.010.0120.0140.0160.0180.02

15°C

148°

C

41.1 41.2 41.32.2

2.4

2.6

2.8

3

3.2

I (m

A)

t (hrs)

Pote

ntia

l (V)

0.002

0.003

0.004

0.005

0.006

Figure 3. Conductor1/ferroelectric-electrolyte/conductor2 cells showing self-charging and self-cycling while the cell isset to discharge. (a) Zn/ferroelectric-electrolyte (Na2.99Ba0.005ClO)/C + Cu cell showing bifurcations corresponding toself-cycling while discharging with a resistor of 1000 Ω at 60 C as in Figure 1e (adapted from [70]); (b) Cu/ferroelectric-electrolyte (Na2.99Ba0.005ClO)/Cu cell after quick charge followed by discharge at open-circuit voltage to 1.1–1.3 V, for thenself-charging to 2.3–2.4 V after 6 or 8 h up to approximately 9 or 5 V, respectively (adapted from [65]).

As in the Poincaré maps where a periodic solution of the feedback system is attained(Figure 2d,e), it is possible to achieve a periodic self-charge/discharge in cells of the typeconductor1/ferroelectric-electrolyte/conductor2, as shown in Figures 3 and 4a–d [65–74].Results obtained with Na2.99Ba0.005ClO and Li2.99Ba0.005ClO ferroelectric-based cells showthat self-charge and self-cycles may be described with van der Pol’s model (Figures 3 and 4).In certain circumstances, the voltage of our cells can be described by Duffing oscillators as,for example, in Figure 4e–h similar to Figure 1g.

This study aimed to observe the relationship between the microscopic and macroscopicproperties of the amorphous simulated Li3ClO, Na3ClO, Li2.92Ba0.04ClO, Na2.92Ba0.04ClOelectrolytes, and to compare them with the amorphous SiO2. Understanding if the lattermay withstand decoherence, or if full-cell emergent phenomena are necessary to observecoherence, was an additional goal. It is noteworthy that the features obtained hereafterin this study for the latter ferroelectric-electrolytes are related to phonons, not with theelectrons. These behaviors are nonetheless linked. As in superconducting topologicalbehavior, the electron-phonon coupling is observed.

To what extent are the configurations of the ferroelectric-electrolyte cells propitiatingcoherent self-oscillations (such as those in Figures 3 and 4)? Is it a ferroelectric-electrolytesolely driven phenomenon? Or it requires a certain cell architecture? What is the roleof the electrolytes’ Barium in decoherence? The goal of this study is to shed light onthese questions.

Materials 2021, 14, 2398 8 of 16

Materials 2021, 14, x 7 of 16

(a) (b)

Figure 3. conductor1/ferroelectric-electrolyte/conductor2 cells showing self-charging and self-cycling while the cell is set to discharge. (a) Zn/ferroelectric-electrolyte (Na2.99Ba0.005ClO)/C + Cu cell showing bifurcations corresponding to self-cy-cling while discharging with a resistor of 1000 Ω at 60 °C as in Figure 1e (adapted from [70]); (b) Cu/ferroelectric-electrolyte (Na2.99Ba0.005ClO)/Cu cell after quick charge followed by discharge at open-circuit voltage to 1.1–1.3 V, for then self-charg-ing to 2.3–2.4 V after 6 or 8 h up to approximately 9 or 5 V, respectively (adapted from [65]).

In our previous studies, it was demonstrated that coherence might be observed while discharging electrostatic/electrochemical cells with a load [65–74]. In other words, the cells of the type conductor1/ferroelectric-electrolyte/conductor2 with collective electrical self-charge and self-cycling phenomena look like a single macroscopic quantum object while discharging with a load, as observed in Figures 3 and 4. The ferroelectric-electrolytes (Na2.99Ba0.005ClO and Li2.99Ba0.005ClO) and their polymer composites have huge dielectric constants such as those observed in Figure 4e–h; the conductor-electrodes in conduc-tor1/ferroelectric-electrolyte/conductor2 do not contain sodium or lithium, in any form, when the cells are assembled. Experiments show that even when discharge is set at rela-tively high currents such as 0.4–0.2 mA cm-2, coherent self-charge and self-cycling can be observed, especially above 35 °C [70–74].

(a) (b)

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

1.2

t (hrs)

Pote

ntia

l (V)

0 2 4 6 8 10 120123456789

1011

burst discharge

10 min. charge at 1.2 mA 1 h charge at 0.50 mA 2 h charge at 0.15 mA

burst discharge

t (hrs)

Po

tent

ial (

V) burst discharge

Discharge at OCV

2.4 V 2.3 V

0

1

2

3

4

5Vmea

IconPo

tent

ial (

V)

t (hrs)

I (m

A)

Imea

2.4 V

0 10 20 30 40 50 60 70-0.00200.0020.0040.0060.0080.010.0120.0140.0160.0180.02

15°C

148°

C41.1 41.2 41.3

2.2

2.4

2.6

2.8

3

3.2

I (m

A)

t (hrs)

Pote

ntia

l (V)

0.002

0.003

0.004

0.005

0.006

Materials 2021, 14, x 8 of 16

(c) (d)

(e) (f)

(g) (h)

Figure 4. conductor1/ferroelectric-electrolyte/conductor2 cells showing self-charging and self-cycling while set to discharge and amplitude jumps (discontinuities) related frequency response in a Duffing oscillator as shown in Figure 1g; (a) self-charge and self-oscillations observed with an Al/90 wt. % Li-glass (Li2.99Ba0.005ClO) + 10 wt. % Li2S/Cu cell while set to discharge at a constant current of −1 µA (adapted from [70]); (b) inset of a) showing small oscillations like n in Figure 2f and w (current) in Figure 2g; (c) and (d) phase portraits of the voltage and current oscillations shown in (a,b)which clearly show limit cycles; (e) and (f) relative real and imaginary permittivities for Na2.99Ba0.005ClO for different times at 25 °C showing jumps in the amplitude of the forced oscillations imposed by the electrical impedance spectroscopy (EIS) meas-urement (adapted from [66]); (g) relative real and imaginary permittivities ratio for Li2.99Ba0.005ClO, at different tempera-tures, showing jumps in the amplitude of the forced oscillations imposed by the electrical impedance spectroscopy (EIS) measurement driving forces; (h) imaginary vs real impedances for Li2.99Ba0.005ClO, at different temperatures, showing jumps in the amplitude of the forced oscillations imposed by the electrical impedance spectroscopy (EIS) (adapted from [64]). Note: EIS is an AC technique, therefore, a frequency response as shown in Figure 1g for a Duffing oscillator is ex-pected. The voltage amplitude was 10 mV.

As in the Poincaré maps where a periodic solution of the feedback system is attained (Figure 2(d,e), it is possible to achieve a periodic self-charge/discharge in cells of the type

2.5 3 3.5

-0.01

0

0.01

dV/d

t (Vs

-1)

Potential V0.002 0.003 0.004 0.005

-3x10-5

-2x10-5

-1x10-5

0

1x10-5

2x10-5

3x10-5

I(mA)

dI/d

t (m

As-1

)

10-2 10-1 100 101 102 103 104 105 106 107 108101

102

103

104

105

106

107

108

109Na3-2x0.005Ba0.005ClO

ε r'

f(Hz)

25 ºC_a 25 ºC_b 25 ºC_c 25 ºC_d 25 ºC_e 25 ºC_f

10-2 10-1 100 101 102 103 104 105 106 107101

102

103

104

105

106

107

108

25 ºC_a 25 ºC_b 25 ºC_c 25 ºC_d 25 ºC_e 25 ºC_f

ε r''

Na3-2x0.005Ba0.005ClO

f(Hz)

-2 -1 0 1 2 3 4 5 6 70

2

4

6

28° C 32° C 50° C 66° C 79° C 87° C 101° C 109° C 117° C 123° C 135° C 141° C 183° C 208° C 216° C 263° C

ε'/ε''

log f(Hz)

Li3-2x0.005Ba0.005ClO

-20 0 20 40 60 80 100 120 140-20

0

20

40

60

80

100

120

140

Z' (Ω)

25ºC 15ºC 10ºC 5ºC 0ºC -5ºC -10ºC -15ºC -20ºC -25ºC -30ºC -35ºC

-Z''

(Ω)

106 Hz

Li3-2x0.005Ba0.005ClO

Figure 4. Conductor1/ferroelectric-electrolyte/conductor2 cells showing self-charging and self-cycling while set to discharge and amplitude jumps (discontinuities) related frequency responsein a Duffing oscillator as shown in Figure 1g; (a) self-charge and self-oscillations observed with anAl/90 wt. % Li-glass (Li2.99Ba0.005ClO) + 10 wt. % Li2S/Cu cell while set to discharge at a constantcurrent of−1 µA (adapted from [70]); (b) inset of (a) showing small oscillations like n in Figure 2f andw (current) in Figure 2g; (c,d) phase portraits of the voltage and current oscillations shown in (a,b)which clearly show limit cycles; (e,f) relative real and imaginary permittivities for Na2.99Ba0.005ClOfor different times at 25 C showing jumps in the amplitude of the forced oscillations imposedby the electrical impedance spectroscopy (EIS) measurement (adapted from [66]); (g) relative realand imaginary permittivities ratio for Li2.99Ba0.005ClO, at different temperatures, showing jumpsin the amplitude of the forced oscillations imposed by the electrical impedance spectroscopy (EIS)measurement driving forces; (h) imaginary vs real impedances for Li2.99Ba0.005ClO, at differenttemperatures, showing jumps in the amplitude of the forced oscillations imposed by the electricalimpedance spectroscopy (EIS) (adapted from [64]). Note: EIS is an AC technique, therefore, afrequency response as shown in Figure 1g for a Duffing oscillator is expected. The voltage amplitudewas 10 mV.

Materials 2021, 14, 2398 9 of 16

3. Theoretical Methods

The relationship between microscopic and macroscopic properties is not intuitive,because fluctuations of properties are significant in microscopic systems. It is required,therefore, to collect multiple snapshots of a microscopic system to compute a meaningfulaverage property. This collection of snapshots is a statistical ensemble.

The molecular dynamics (MD) method enables one to build of a model in whichthe molecules are continuously moving. It simulates the behavior of a limited numberof molecules, confined to a given volume and interacting with one another through agiven pair potential [75]. The molecules are endowed with more or less ordered positioncoordinates at the start, so that none overlap, but with random coordinates of velocity.Their trajectories will then be followed, by a detailed solution of Newton’s laws of motion.

In an ab-initio molecular dynamics (AMD) run the forces calculated in a given ge-ometry step are used to update the atomic positions. The system dynamics, i.e., the ionicmovements, are subject to Newton mechanics while the forces acting on the ions are calcu-lated from ab initio using a self-consistent electronic density (Hellmann-Feynman forces)as implemented in Vienna Ab Initio Software Package (VASP) [76].

Periodic boundary conditions are assumed and allow that a molecule, which leavesthe volume across one face, re-enters across the opposite phase.

In the statistical ensemble, the various states of the system differ in positions and ve-locities of the component’s particles. The space of all possible system states is of dimension6N for N particles and is termed the phase space.

A system at imposed volume V’, and temperature T, is represented by a statisticalensemble, which is called the canonical ensemble or NV′T ensemble.

The number of molecules and the global volume are identical for all system statesbelonging to the ensemble, but they differ in total energy, which is a fluctuating variable inthis ensemble. Each state j of the canonical ensemble occurs with a probability proportionalto e−Ej/kBT where kB is the Boltzmann and Ej is the total energy (kinetic + potential) of thesystem in state j. Here we use the symbol E for total energy (kinetic and potential) and Ufor the potential energy. The Boltzmann factor e−Ej/kBT expresses that low energy statesare favored when compared to high-energy states. Increasing temperature broadens theenergy distribution in the ensemble and, as a consequence, the average energy is increased.

Ab initio molecular dynamics (AMD) as implemented by VASP [76] was used tosimulate a closed system thermostatted in a heat bath at a constant temperature as inan NV′T ensemble. The studied systems were Li81Cl27O27, Li79BaCl27O27, Na81Cl27O27,Na79BaCl27O27, Si3O6, and Si108O216 where the compositions of the compounds contain-ing barium, were the most approximate to A2.99Ba0.005ClO (A = Li or Na), while stillmaintaining a feasible simulating computing time.

The initial structures were optimized at the correspondent temperatures after per-forming microcanonical simulation NV’E and isothermal-isobaric simulations NP’T on thecrystalline optimized structure to set the volumes at the correspondent temperatures. Thetemperatures were chosen to assure that the system is in the amorphous state, immediatelyabove the transition to amorphous.

Projector augmented wave (PAW) potentials [77] in the generalized gradient approx-imation (GGA) [78] were used. Electronic structure calculations were conducted withcut-off energies of 500 eV. The AMD simulations were performed with 4 fs time-steps to amaximum of 8 ps; velocities were rescaled to maintain a constant temperature using theNosé’s thermostat [79] with a mass chosen such that temperature fluctuates with a periodof 40 time-steps. Each NV’T ensemble was allowed to relax at least until decoherence wasnoticeably observed. Results are shown in Figures 5–8.

A finite collection of system states, which is representative of the NV′T statisticalensemble, average properties was obtained by simple arithmetic averaging over the nstates composing the collection. For instance, the average energy is 〈E〉 = 1

n ∑ni=1 Ei. The

heat capacity at constant volume is CV′ =(

∂E∂T

)V′

= 1kBT2

(⟨E2⟩− 〈E〉2).

Materials 2021, 14, 2398 10 of 16

The pair distribution functions g(r) versus the interatomic distances r were alsocalculated for controlling the non-periodic character of the amorphous structures. Anexample of the latter is shown for Li2.92Ba0.04ClO in Figure 5b.

It is important highlighting that if a manifestation is observed during the AMDrelaxation, the correspondent event should be witnessed in Nature. Nonetheless, processeswith slow dynamics will not be observed in AMD, even if occurring in Nature.

The x vs time series and the phase portrait (projection of the extended 3D portrait) forvalues A =−50, ε = 5, andω = 7, corresponding to almost periodic orbit in Figure 2d were ob-tained using wxMaxima 20.06.6 and the Runge-Kutta method in supplementary information.

Materials 2021, 14, x 10 of 16

most approximate to A2.99Ba0.005ClO (A = Li or Na), while still maintaining a feasible sim-ulating computing time.

The initial structures were optimized at the correspondent temperatures after per-forming microcanonical simulation NV’E and isothermal-isobaric simulations NP’T on the crystalline optimized structure to set the volumes at the correspondent temperatures. The temperatures were chosen to assure that the system is in the amorphous state, immedi-ately above the transition to amorphous.

Projector augmented wave (PAW) potentials [77] in the generalized gradient approx-imation (GGA) [78] were used. Electronic structure calculations were conducted with cut-off energies of 500 eV. The AMD simulations were performed with 4 fs time-steps to a maximum of 8 ps; velocities were rescaled to maintain a constant temperature using the Nosé’s thermostat [79] with a mass chosen such that temperature fluctuates with a period of 40 time-steps. Each NV’T ensemble was allowed to relax at least until decoherence was noticeably observed. Results are shown in Figures 5–8.

(a) (b)

(c) (d)

Figure 5. Ab Initio Molecular Dynamics for Li-glass ferroelectric-electrolytes; (a)potential energy vs time for Li2.92Ba0.04ClO and Li3ClO. The upper envelope energy for Li2.92Ba0.04ClO at 363 K before decoherence: −2085 + 41.94𝑒 / . kJ/molfor-

mula unit corresponding to a damping oscillator; decoherence is observed above 1.2 ps; the calculated specific heat is Cv’ = 0.193 kJ/kg K; the upper envelope energy for Li3ClO at 473 K before decoherence: −2085 + 30.23𝑒 / . kJ/molformula unit corresponding to a damping oscillator; decoherence is observed above 7 ps; the calculated specific heat is Cv’ = 0.465 kJ/kg K; (b) pair correlation function for Li2.92Ba0.04ClO, at 363 K, showing first and second neighbors at fixed distances corre-sponding to sharp peaks and broad peaks for third and above neighbors’ distances indicating structural disorder, often associated with an amorphous character, as expected from experimental results; (c)phase portrait for power vs energy for

0 0.8 1.6 2.4 3.2 4 4.8 5.6 6.4 7.2 8

-2120

-2100

-2080

-2060

-2040

Li3-2×0.04Ba0.04ClO (363 K) Li3ClO (473 K)

T

U (k

J/m

olfo

rmul

a un

it)

Tinitial = 0.26 ps, f = 3.8 × 1012 Hz

Tinitial = 0.24 ps, f = 4.2 × 1012 Hz

t (ps)

<U> = -2089.4 kJ/mol

<U> = -2068.8 kJ/mol

0 2 4 6 8 100

0.5

1

1.5

2

2.5

3

Pair

cor

rela

tion

func

tion

distance (Å)

-2110 -2100 -2090 -2080 -2070 -2060-0.2

-0.1

0

0.1

0.2Li3ClO (473 K)

P (k

J/fs

.mol

form

ula

unit)

U (kJ/molformula unit)-2080 -2070 -2060 -2050

-0.4

-0.2

0

0.2

0.4

Li3-2×0.04Ba0.04ClO (363 K)

P (k

J/s.

mol

form

ula

unit)

U (kJ/molformula unit)

Figure 5. Ab Initio Molecular Dynamics for Li-glass ferroelectric-electrolytes; (a) potential energy vs time for Li2.92Ba0.04ClOand Li3ClO. The upper envelope energy for Li2.92Ba0.04ClO at 363 K before decoherence: −2085 + 41.94e−t/5.634 kJ/molformula unit corresponding to a damping oscillator; decoherence is observed above 1.2 ps; the calculated specific heat is Cv’

= 0.193 kJ/kg K; the upper envelope energy for Li3ClO at 473 K before decoherence:−2085+ 30.23e−t/2.012 kJ/molformula unit

corresponding to a damping oscillator; decoherence is observed above 7 ps; the calculated specific heat is Cv’ = 0.465 kJ/kg K;(b) pair correlation function for Li2.92Ba0.04ClO, at 363 K, showing first and second neighbors at fixed distances correspondingto sharp peaks and broad peaks for third and above neighbors’ distances indicating structural disorder, often associatedwith an amorphous character, as expected from experimental results; (c) phase portrait for power vs. energy for Li3ClO at473 K corresponding to a damped coherent oscillator that becomes decoherent; the relaxation of the focus (equilibriumenergy) corresponds to a variation of 0.3% from the initial focus energy; ∆Umax = Umax − Umin ≈ 0.46 eV; (d) phase portraitfor power vs energy for Li2.92Ba0.04ClO, at 363 K, corresponding to a damped coherent oscillator that becomes decoherent;the relaxation of the focus corresponds to a variation of 0.2% of the initial energy; ∆Umax ≈ 0.32 eV. Note: both (b) and (c)are similar to the Duffing oscillator model in Figure 1f.

Materials 2021, 14, 2398 11 of 16

Materials 2021, 14, x 11 of 16

Li3ClO at 473 K corresponding to a damped coherent oscillator that becomes decoherent; the relaxation of the focus (equi-librium energy) corresponds to a variation of 0.3% from the initial focus energy; ΔUmax = Umax − Umin ≈ 0.46 eV; (d) phase portrait for power vs energy for Li2.92Ba0.04ClO, at 363 K, corresponding to a damped coherent oscillator that becomes decoherent; the relaxation of the focus corresponds to a variation of 0.2% of the initial energy; ΔUmax ≈ 0.32 eV. Note: both (b) and (c)are similar to the Duffing oscillator model in Figure 1f.

(a) (b)

(c) (d)

Figure 6. Ab Initio Molecular Dynamics for Li-glass ferroelectric-electrolytes showing decoherence; (a) inset of the phase portrait of Li3ClO showing a double-well potential; (b) schematic representation of energy vs position, showing that ther-mal excitation allows a molecule to hop back and forth between the wells; (c) and (d) insets of the phase portrait of Li2.92Ba0.04ClO showing back and forth paths between the wells described by a van der Pol-Duffing oscillator model as the one in Figs. 1f and 2d. Note: Potential energy is 𝑑𝑈 = −𝑭. 𝒅 and power is 𝑑𝑃 = 𝑭. 𝒅𝒗 where 𝑭 is an applied force, 𝒓 atom displacement, and 𝒗 atom velocity.

(a) (b)

-2095 -2090 -2085-0.05-0.04-0.03-0.02-0.01

00.010.020.030.040.05

Li3ClO (473 K)

kBT = 0.0408 eV

P (k

J/(fs

.mol

form

ula

unit)

U (kJ/molformula unit)

-21.6199 eV

-21.7034 eV

-21.6785 eV-21.6749 eV

-2072 -2070 -2068 -2066

-0.1

0

0.1

Li3-2×0.04Ba0.04ClO

P (k

J/s.

mol

form

ula

unit)

U (kJ/molformula unit)

T = 363 KkBT = 0.0313 eV

0.0166 eV

-2070 -2069 -2068-0.05

0

0.05

0.0313 eV

1

2

3

4 5

6

U (kJ/molformula unit)

Li3-2×0.04Ba0.04ClO

T = 363 KkBT = 0.0313 eV

P (k

J/s.

mol

form

ula

unit)

0 200 400 600 800 1000 1200 1400

-1730-1720-1710-1700-1690-1680-1670-1660

U (k

J/m

olfo

rmul

a un

it)

t (fs)

Na3-2×0.04Ba0.04ClO (310 K)

<U> = -1677.6 kJ/mol

Na3ClO (310 K)

<U> = -1714.9 kJ/mol

-1700 -1690 -1680 -1670 -1660 -1650-2

-1

0

1

2

U (kJ/molformula unit)

P (k

J/fs

.mol

form

ula

unit)

Na3-2×0.04Ba0.04ClO (310 K)

Figure 6. Ab Initio Molecular Dynamics for Li-glass ferroelectric-electrolytes showing decoherence;(a) inset of the phase portrait of Li3ClO showing a double-well potential; (b) schematic representationof energy vs position, showing that thermal excitation allows a molecule to hop back and forthbetween the wells; (c,d) insets of the phase portrait of Li2.92Ba0.04ClO showing back and forth pathsbetween the wells described by a van der Pol-Duffing oscillator model as the one in Figures 1f and 2d.Note: Potential energy is dU = −F.dr and power is dP = F.dv where F is an applied force, r atomdisplacement, and v atom velocity.

Materials 2021, 14, x 11 of 16

Li3ClO at 473 K corresponding to a damped coherent oscillator that becomes decoherent; the relaxation of the focus (equi-librium energy) corresponds to a variation of 0.3% from the initial focus energy; ΔUmax = Umax − Umin ≈ 0.46 eV; (d) phase portrait for power vs energy for Li2.92Ba0.04ClO, at 363 K, corresponding to a damped coherent oscillator that becomes decoherent; the relaxation of the focus corresponds to a variation of 0.2% of the initial energy; ΔUmax ≈ 0.32 eV. Note: both (b) and (c)are similar to the Duffing oscillator model in Figure 1f.

(a) (b)

(c) (d)

Figure 6. Ab Initio Molecular Dynamics for Li-glass ferroelectric-electrolytes showing decoherence; (a) inset of the phase portrait of Li3ClO showing a double-well potential; (b) schematic representation of energy vs position, showing that ther-mal excitation allows a molecule to hop back and forth between the wells; (c) and (d) insets of the phase portrait of Li2.92Ba0.04ClO showing back and forth paths between the wells described by a van der Pol-Duffing oscillator model as the one in Figs. 1f and 2d. Note: Potential energy is 𝑑𝑈 = −𝑭. 𝒅 and power is 𝑑𝑃 = 𝑭. 𝒅𝒗 where 𝑭 is an applied force, 𝒓 atom displacement, and 𝒗 atom velocity.

(a) (b)

-2095 -2090 -2085-0.05-0.04-0.03-0.02-0.01

00.010.020.030.040.05

Li3ClO (473 K)

kBT = 0.0408 eV

P (k

J/(fs

.mol

form

ula

unit)

U (kJ/molformula unit)

-21.6199 eV

-21.7034 eV

-21.6785 eV-21.6749 eV

-2072 -2070 -2068 -2066

-0.1

0

0.1

Li3-2×0.04Ba0.04ClO

P (k

J/s.

mol

form

ula

unit)

U (kJ/molformula unit)

T = 363 KkBT = 0.0313 eV

0.0166 eV

-2070 -2069 -2068-0.05

0

0.05

0.0313 eV

1

2

3

4 5

6

U (kJ/molformula unit)

Li3-2×0.04Ba0.04ClO

T = 363 KkBT = 0.0313 eV

P (k

J/s.

mol

form

ula

unit)

0 200 400 600 800 1000 1200 1400

-1730-1720-1710-1700-1690-1680-1670-1660

U (k

J/m

olfo

rmul

a un

it)

t (fs)

Na3-2×0.04Ba0.04ClO (310 K)

<U> = -1677.6 kJ/mol

Na3ClO (310 K)

<U> = -1714.9 kJ/mol

-1700 -1690 -1680 -1670 -1660 -1650-2

-1

0

1

2

U (kJ/molformula unit)

P (k

J/fs

.mol

form

ula

unit)

Na3-2×0.04Ba0.04ClO (310 K)

Figure 7. Ab Initio Molecular Dynamics for Na-glass ferroelectric-electrolyte showing decoher-ence; (a) potential energy vs time for Na2.92Ba0.04ClO at 310 K; the calculated specific heat isCv’ = 0.442 kJ/kg K and potential energy vs time for Na3ClO at 310 K; the calculated specific heatis Cv’ = 0.457 kJ/kg K; (b) phase portrait of Na2.92Ba0.04ClO for power vs potential energy showingback and forth between two double-wells described by a van der Pol-Duffing oscillator model as theone in Figure 1f.

Materials 2021, 14, 2398 12 of 16

Materials 2021, 14, x 12 of 16

Figure 7. Ab Initio Molecular Dynamics for Na-glass ferroelectric-electrolyte showing decoherence; (a) potential energy vs time for Na2.92Ba0.04ClO at 310 K; the calculated specific heat is Cv’ = 0.442 kJ/kg K and potential energy vs time for Na3ClO at 310 K; the calculated specific heat is Cv’ = 0.457 kJ/kg K; (b) phase portrait of Na2.92Ba0.04ClO for power vs poten-tial energy showing back and forth between two double-wells described by a van der Pol-Duffing oscillator model as the one in Figure 1f.

(a) (b)

Figure 8. Ab Initio Molecular Dynamics for the amorphous Si3O6 showing noise and a quasi-periodic structure; (a) total energy vs time for SiO2 at 1600 K; the calculated specific heat is Cv’ = 0.826 kJ/kg K; no damping is observed; (b) phase portrait of SiO2 for power vs potential energy showing one energy well and almost no anharmonicity. The difference between the two energy states is ΔU ≈ 0.62 eV and KBT = 0.129 eV. Almost no focus (equilibrium energy) displacement is observed. Note that simulations not shown here demonstrate that for a larger box such as that containing Si108O216, deco-herence is achieved immediately even at room temperature.

A finite collection of system states, which is representative of the 𝑁𝑉’𝑇 statistical en-semble, average properties was obtained by simple arithmetic averaging over the 𝑛 states composing the collection. For instance, the average energy is < 𝐸 >= ∑ 𝐸 . The heat

capacity at constant volume is 𝐶 = = (< 𝐸 > − < 𝐸 > ). The pair distribution functions 𝑔(𝑟) versus the interatomic distances 𝑟 were also

calculated for controlling the non-periodic character of the amorphous structures. An ex-ample of the latter is shown for Li2.92Ba0.04ClO in Figure 5b.

It is important highlighting that if a manifestation is observed during the AMD re-laxation, the correspondent event should be witnessed in Nature. Nonetheless, processes with slow dynamics will not be observed in AMD, even if occurring in Nature.

The x vs time series and the phase portrait (projection of the extended 3D portrait) for values A = −50, ε = 5, and ω = 7, corresponding to almost periodic orbit in Figure 2d were obtained using wxMaxima 20.06.6 and the Runge-Kutta method in supplementary information.

4. Results and Discussion The results obtained for the ferroelectric amorphous-electrolytes have been demon-

strated to be very different from those obtained for the amorphous SiO2 that was used as a control, as its heat capacity is well known (Figures 5–8).

While Li3ClO phonons’ oscillations corresponded to a strong environmental coupling between 0 and 7.0 ps, Li2.92Ba0.04ClO phonon oscillations were only strongly coupled up to 1.2 ps (Figure 5). Conversely, Si3O6 shows an almost periodic orbit (Figure 2d) where noise is observed since 𝑡 = 0 ps. As expected, Na3ClO maintains a certain coherence during the first ps, while Na2.92Ba0.04ClO phonon oscillations show decoherence almost immediately (Figure 7).

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-2280-2260-2240-2220-2200-2180-2160-2140-2120-2100 SiO2 (1600 K), amorphous

t (ps)

U (k

J/m

olfo

rmul

a un

it)

<U> = -2201.7 kJ/mol

-2240 -2220 -2200 -2180 -2160-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

U (kJ/molformula unit)

P (k

J/fs

.mol

form

ula

unit)

SiO2 (1600 K)

Commented [M61]: Please check if this is a citation of Supplementary Materials. If so, please check if there is a Supplementary Materials section in Back Matter of this manuscript

Figure 8. Ab Initio Molecular Dynamics for the amorphous Si3O6 showing noise and a quasi-periodicstructure; (a) total energy vs time for SiO2 at 1600 K; the calculated specific heat is Cv’ = 0.826 kJ/kg K;no damping is observed; (b) phase portrait of SiO2 for power vs potential energy showing one energywell and almost no anharmonicity. The difference between the two energy states is ∆U ≈ 0.62 eVand kBT = 0.129 eV. Almost no focus (equilibrium energy) displacement is observed. Note thatsimulations not shown here demonstrate that for a larger box such as that containing Si108O216,decoherence is achieved immediately even at room temperature.

4. Results and Discussion

The results obtained for the ferroelectric amorphous-electrolytes have been demon-strated to be very different from those obtained for the amorphous SiO2 that was used as acontrol, as its heat capacity is well known (Figures 5–8).

While Li3ClO phonons’ oscillations corresponded to a strong environmental couplingbetween 0 and 7.0 ps, Li2.92Ba0.04ClO phonon oscillations were only strongly coupled up to1.2 ps (Figure 5). Conversely, Si3O6 shows an almost periodic orbit (Figure 2d) where noiseis observed since t = 0 ps. As expected, Na3ClO maintains a certain coherence during thefirst ps, while Na2.92Ba0.04ClO phonon oscillations show decoherence almost immediately(Figure 7).

Whereas Li3ClO and Li2.92Ba0.04ClO show single-well oscillations in the first ps(Figures 5 and 6), Na3ClO and Na2.92Ba0.04ClO show almost instantaneous decoherencewhile their energy oscillates in a double-well (Figure 7). Si3O6 energy oscillates in a singlewell and shows noise since the first time-steps (Figure 8). Control simulations performedwith Si108O216 show that SiO2 phonon oscillations are immediately decoherent (not show-ing any periodicity), even at room temperature, which indicates that if the simulationbox contains enough atoms, the decoherence is obtained faster. It is not trivial to obtain acanonical ensemble distribution of conformations and velocities as they depend on systemsize, thermostat, and time step.

Although decoherence is no doubtfully more prominent at elevated temperaturesdue to its very nature, in this study it was observed that the substitution of two Lithiumatoms by a heavy Barium atom and a vacancy is more important, surpassing the effect oftemperature as observed in Figure 5; Li3ClO keeps its coherence for a much longer periodthan Li2.92Ba0.04ClO, even if its temperature is 110 K higher. The same applies to the Na-based ferroelectric electrolyte as shown later. We had previously observed experimentallythat the introduction of a small amount of Ba in Li2.99Ba0.005ClO reduces the glass transitiontemperature of Li3ClO.

Two very distinctive features are found when comparing the amorphous ferroelectricelectrolytes and the amorphous Si3O6 AMD relaxation oscillations; while the first is dissi-pative and anharmonic (Figure 5), in the second the dissipation is not observed and thesystem keeps its focal energy point fairly well; no exponential dissipation envelop-typecould be observed in Figure 8 for SiO2 at 1600 K. This phenomenon does not seem to bedirectly related with the temperature difference as simulations for the latter at 298 K showcomparable results.

Materials 2021, 14, 2398 13 of 16

The upper envelope of the potential energy for Li2.92Ba0.04ClO, at 363 K, before deco-herence, is −2085 + 41.94e-t/5.634 kJ/molformula unit corresponding to a damping oscillator.Decoherence is observed above 1.2 ps; the calculated specific heat at constant volume isCv’ = 0.193 kJ/kg K. The upper envelope energy for Li3ClO at 473 K before decoherence is−2085 + 30.23e-t/2.012 kJ/molformula unit corresponding to a damping oscillator; decoherenceis observed above 7.0 ps (Figure 5). The calculated specific heat at constant volume isCv’ = 0.465 kJ/kg K. The difference between envelope curves should be essentially relatedto Ba doping.

The total energy vs time for Na2.92Ba0.04ClO, at 310 K, shows almost instantaneous de-coherence; the calculated specific heat at constant volume is Cv’ = 0.442 kJ/kg K (Figure 7).For Na3ClO at 310 K the decoherence occurs at approximately 1 ps and the calculatedspecific heat at constant volume is Cv’ = 0.457 kJ/kg K (Figure 7). For SiO2 at 1600 K, thecalculated specific heat at constant volume is Cv’ = 0.826 kJ/kg K (Figure 8).

5. Conclusions

The ab initio molecular dynamics canonical ensemble was used to describe the ther-modynamics of five different amorphous systems (Li3ClO, Li3−2 × 0.04Ba0.04ClO, Na3ClO,Na3-2 × 0.04Ba0.04ClO, and SiO2) as the systems were thermostatted in a heat bath at aconstant temperature. This procedure allowed calculating the potential energy and theaverage of the total energy of each system, as well as, the specific heat at constant volume.The results for Li3ClO, Li3−2 × 0.04Ba0.04ClO, Na3ClO, and Na3−2 × 0.04Ba0.04ClO specificheat at constant volume are in agreement with the experimental results of the end binaries;for SiO2 the calculated specific heat is within the experimental error of data found inthe literature.

The ferroelectric nature of the Li3ClO, Li3−2 × 0.04Ba0.04ClO, Na3ClO, andNa3−2 × 0.04Ba0.04ClO was observed as the potential energy relaxed to two energy minimaaccessible by thermal energy, kBT, corresponding to symmetric configurations of dipoles ordipoles coalescences as in an Ising model. These systems tend to be described by van derPol– Duffing oscillators.

In this study, it was observed that the substitution of two lithium or sodium atoms by aheavy barium and a vacancy is more important than the effect of temperature in decoherence.

Although the ferroelectric character of the electrolytes, the van der Pol–Duffingrelaxation-oscillator models and a certain coherence are undoubtedly observed for the sim-ulated electrolytes, at the temperatures at which the conductor1/ferroelectric-electrolyte/conductor2 cells were observed to self-charge and self-cycle; the time-scales for coherenceobservation are much smaller. Therefore, it is likely that a cell containing two electrodeswith different electrochemical potentials and the electrolyte is needed to have relaxationoscillations in a timescale of hours or days confirming the phenomenon as emergent.The external circuit discharge with the internal self-charge feedback enabled by the cou-pled electron-phonon surface conduction of the ferroelectric-electrolyte constitutes twoPoincaré maps. This latter feedback described by equivalent circuits such as van der Poland Hodgkin-Huxley FitzHugh-Nagumo is likely to constitute a necessary condition toachieve coherence in a ferroelectric-electrolyte cell.

Supplementary Materials: The following are available online at https://www.mdpi.com/article/10.3390/ma14092398/s1.

Funding: The author is grateful for John B. Goodenough’s (Nobel Prize of Chemistry 2019) endow-ment and the support of the Portuguese Foundation for Science and Technology under the FCT grantfor the UIDP/50022/2020 LAETA project.

Institutional Review Board Statement: Not applicable.

Informed Consent Statement: Not applicable.

Data Availability Statement: The data is available upon demand.

Conflicts of Interest: The author declares no conflict of interest.

Materials 2021, 14, 2398 14 of 16

References1. Noé, F.; Olsson, S.; Köhler, J.; Wu, H. Boltzmann generators: Sampling equilibrium states of many-body systems with deep

learning. Science 2019, 365, eaaw1147. [CrossRef] [PubMed]2. O’Connell, A.D.; Hofheinz, M.; Ansmann, M.; Bialczak, R.C.; Lenander, M.; Lucero, E.; Neeley, M.; Sank, D.; Wang, H.; Weides,

M.; et al. Quantum ground state and single-phonon control of a mechanical resonator. Nature 2010, 464, 697–703. [CrossRef]3. Sachdev, S. Quantum Phase Transitions; Cambridge University Press: Cambridge, UK, 1999.4. Tanaka, S.; Tamura, R.; Chakrabarti, B.K. Quantum Spin Glasses, Annealing and Computation; Cambridge University Press:

Cambridge, UK, 2007.5. Classical view of the properties of Rydberg atoms: Application of the correspondence principle. Am. J. Phys. 1992, 60, 329–335.

[CrossRef]6. Schauss, P. Quantum simulation of transverse Ising models with Rydberg atoms. Quantum Sci. Technol. 2018, 3, 023001. [CrossRef]7. Auerbach, A. Interacting Electrons and Quantum Magnetism; Springer: New York, NY, USA, 1994.8. Schollwöck, U.; Richter, J.; Farnell, D.; Bishop, R. (Eds.) Quantum Magnetism; (Lect. Notes Phys.); Springer: Berlin, Germany, 2004;

p. 645.9. Parkinson, J.; Farnell, D. An Introduction to Quantum Spin Systems; (Lect. Notes Phys); Springer: Berlin, Germany, 2010; p. 816.10. Johnson, M.W.; Amin, M.H.S.; Gildert, S.; Lanting, T.; Hamze, F.; Dickson, N.G.; Harris, R.M.; Berkley, A.J.; Johansson, J.; Bunyk,

P.; et al. Quantum annealing with manufactured spins. Nature 2011, 473, 194. [CrossRef]11. Rønnow, T.F.; Wang, Z.; Job, J.; Boixo, S.; Isakov, S.V.; Wecker, D.; Martinis, J.M.; Lidar, D.A.; Troyer, M. Defining and detecting

quantum speedup. Science 2014, 345, 420. [CrossRef]12. Glaetzle, A.W.; Van Bijnen, R.M.W.; Zoller, P.; Lechner, W. A coherent quantum annealer with Rydberg atoms. Nat. Commun.

2017, 8, 15813. [CrossRef]13. Yoshihara, F.; Nakamura, Y.; Tsai, J.S. Correlated flux noise and decoherence in two inductively coupled flux qubits. Phys. Rev. B

2010, 81, 132502. [CrossRef]14. Merlin, R. Rabi oscillations, Floquet states, Fermi’s golden rule, and all that: Insights from an exactly solvable two-level model.

Am. J. Phys. 2021, 89, 26. [CrossRef]15. Shore, B.W.; Knight, P.L. The Jaynes-Cummings Model. J. Mod. Opt. 1993, 40, 1195–1238. [CrossRef]16. Knight, P.L.; Milonni, P.W. The Rabi frequency in optical spectra. Phys. Rep. 1980, 66, 21–107. [CrossRef]17. Dudin, Y.O.; Li, L.; Bariani, F.; Kuzmich, A. Observation of coherent many-body Rabi oscillations. Nat. Phys. 2012, 12, 790–794.

[CrossRef]18. Pileio, G.; Carravetta, M.; Levitt, M.H. Extremely Low-Frequency Spectroscopy in Low-Field Nuclear Magnetic Resonance. Phys.

Rev. Lett. 2009, 103, 083002. [CrossRef] [PubMed]19. Houdré, R.; Stanley, R.P.; Oesterle, U.; Ilegems, M.; Weisbuch, C. Room-temperature cavity polaritons in a semiconductor

microcavity. Phys. Rev. B 1994, 49, 16761–16764. [CrossRef]20. Deng, H.; Haug, H.; Yamamoto, Y. Exciton-polariton Bose-Einstein condensation. Rev. Mod. Phys. 2010, 82, 1489–1537. [CrossRef]21. Amo, A.; Lefrère, J.; Pigeon, S.; Adrados, C.; Ciuti, C.; Carusotto, I.; Houdré, R.; Giacobino, E.; Bramati, A. Superfluidity of

polaritons in semiconductor microcavities. Nat. Phys. 2009, 5, 805–810. [CrossRef]22. Batalov, A.; Zierl, C.; Gaebel, T.; Neumann, P.; Chan, I.-Y.; Balasubramanian, G.; Hemmer, P.R.; Jelezko, F.; Wrachtrup, J. Temporal

Coherence of Photons Emitted by Single Nitrogen-Vacancy Defect Centers in Diamond Using Optical Rabi-Oscillations. Phys.Rev. Lett. 2008, 100, 077401. [CrossRef]

23. Stievater, T.H.; Li, X.; Steel, D.G.; Gammon, D.; Katzer, D.S.; Park, D.; Piermarocchi, C.; Sham, L.J. Rabi Oscillations of Excitons inSingle Quantum Dots. Phys. Rev. Lett. 2001, 87, 133603. [CrossRef]

24. Petta, J.R.; Johnson, A.C.; Taylor, J.M.; Laird, E.A.; Yacoby, A.; Lukin, M.D.; Marcus, C.M.; Hanson, M.P.; Gossard, A.C. Coherentmanipulation of coupled electron spins in semiconductor quantum dots. Science 2005, 309, 2180–2184. [CrossRef]

25. Koppens, F.H.L.; Buizert, C.; Tielrooij, K.J.; Vink, I.T.; Nowack, K.C.; Meunier, T.; Kouwenhoven, L.P.; Vandersypen, L.M.K.Driven coherent oscillations of a single electron spin in a quantum dot. Nature 2006, 442, 766–771. [CrossRef]

26. Martinis, J.M.; Nam, S.; Aumentado, J.; Urbina, C. Rabi Oscillations in a Large Josephson-Junction Qubit. Phys. Rev. Lett. 2002,89, 117901. [CrossRef]

27. Wang, Y.H.; Steinberg, H.; Jarilloherrero, P.; Gedik, N. Observation of Floquet-Bloch States on the Surface of a TopologicalInsulator. Science 2013, 342, 453–457. [CrossRef] [PubMed]

28. Oka, T.; Aoki, H. Photovoltaic Hall effect in graphene. Phys. Rev. B 2009, 79, 081406. [CrossRef]29. Morimoto, T.; Nagaosa, N. Topological nature of nonlinear optical effects in solids. Sci. Adv. 2016, 2, 1501524. [CrossRef]

[PubMed]30. Rechtsman, M.C.; Zeuner, J.M.; Plotnik, Y.; Lumer, Y.; Podolsky, D.K.; Dreisow, F.; Nolte, S.; Segev, M.; Szameit, A. Photonic

Floquet topological insulators. Nature 2013, 496, 196–200. [CrossRef]31. Lindner, N.H.; Refael, G.; Galitski, V. Floquet topological insulator in semiconductor quantum wells. Nat. Phys. 2011, 7, 490–495.

[CrossRef]32. Sentef, M.; Claassen, M.; Kemper, A.F.; Moritz, B.; Oka, T.; Freericks, J.K.; Devereaux, T.P. Theory of Floquet band formation and

local pseudospin textures in pump-probe photoemission of graphene. Nat. Commun. 2015, 6, 7047. [CrossRef]

Materials 2021, 14, 2398 15 of 16

33. Din, Q.; Elsadany, A.A.; Khalil, H. Neimark-Sacker Bifurcation and Chaos Control in a Fractional-Order Plant-Herbivore Model.Discret. Dyn. Nat. Soc. 2017, 2017, 6312964. [CrossRef]

34. Kim, J.; Lee, H.; Shin, J. Extended framework of Hamilton’s principle applied to Duffing oscillation. Appl. Math. Comput. 2020,367, 124762. [CrossRef]

35. Brennan, M.; Kovacic, I.; Carrella, A.; Waters, T. On the jump-up and jump-down frequencies of the Duffing oscillator. J. SoundVib. 2008, 318, 1250–1261. [CrossRef]

36. Gao, J.; Delos, J.B. Quantum manifestations of bifurcations of closed orbits in the photoabsorption spectra of atoms in electricfields. Phys. Rev. A 1997, 56, 356–364. [CrossRef]

37. Peters, A.D.; Jaffé, C.; Delos, J.B. Quantum Manifestations of Bifurcations of Classical Orbits: An Exactly Solvable Model. Phys.Rev. Lett. 1994, 73, 2825–2828. [CrossRef]

38. Courtney, M.; Jiao, H.; Spellmeyer, N.; Kleppner, D.; Gao, J.; Delos, J.B. Closed Orbit Bifurcations in Continuum Stark Spectra.Phys. Rev. Lett. 1995, 74, 1538–1541. [CrossRef]

39. Founargiotakis, M.; Farantos, S.C.; Skokos, C.; Contopoulos, G. Bifurcation diagrams of periodic orbits for unbound molecularsystems: FH2. Chem. Phys. Lett. 1997, 277, 456–464. [CrossRef]

40. Monteiro, T.S.; Saraga, D. Quantum Wells in Tilted Fields: Semiclassical Amplitudes and Phase Coherence Times. Found. Phys.2001, 31, 355–370. [CrossRef]

41. Wieczorek, S.; Krauskopf, B.; Simpson, T.; Lenstra, D. The dynamical complexity of optically injected semiconductor lasers. Phys.Rep. 2005, 416, 1–128. [CrossRef]

42. Stamatiou, G.; Ghikas, D.P. Quantum entanglement dependence on bifurcations and scars in non-autonomous systems. The caseof quantum kicked top. Phys. Lett. A 2007, 368, 206–214. [CrossRef]

43. Galan, J.; Freire, E. Chaos in a mean field model of coupled quantum wells; bifurcations of periodic orbits in a symmetrichamiltonian system. Rep. Math. Phys. 1999, 44, 87–94. [CrossRef]

44. Kleppner, D.; Delos, J.B. Beyond Quantum Mechanics: Insights from the Work of Martin Gutzwiller. Found. Phys. 2001, 31,593–612. [CrossRef]

45. Martin, C.G. Chaos in Classical and Quantum Mechanics; Springer: New York, NY, USA, 1990; ISBN 978-0-387-97173-5.46. Zhang, Y.; Mao, H.; Huang, Z. Detection the nonlinear ultrasonic signals based on modified Duffing equations. Results Phys. 2017,

7, 3243–3250. [CrossRef]47. Varigonda, S.; Georgiou, T. Dynamics of relay relaxation oscillators. IEEE Trans. Autom. Control. Inst. Electr. Electron. Eng. 2001,

46, 65. [CrossRef]48. Du, K.-L.; Swamy, M.N.S. Wireless Communication Systems: From RF Subsystems to 4G Enabling Technologies; Cambridge University

Press: Cambridge, UK, 2010; p. 443. ISBN 1139485768.49. Nave, C.R. Relaxation Oscillator Concept; HyperPhysics, Department of Physics and Astronomy, Georgia State University:

Georgia, GA, USA, 1995. Available online: http://hyperphysics.phy-astr.gsu.edu/Hbase/electronic/relaxo.html (accessed on30 April 2021).

50. Edson, W.A. Vacuum Tube Oscillators; John Wiley and Sons: New York, NY, USA, 1953; p. 3.51. Tsatsos, M. Theoretical and Numerical Study of the Van der Pol Equation. Ph.D. Thesis, Aristotle University of Thessaloniki,

Thessaloniki, Greece, 2006. Available online: https://arxiv.org/ftp/arxiv/papers/0803/0803.1658.pdf (accessed on 30 April 2021).52. Van der Pol, B.; van der Mark, J. Frequency demultiplication. Nature 1927, 120, 363–364. [CrossRef]53. Ebeling, W. Strukturbildung bei Irreversiblen Prozessen; Teubner: Leipzig, Germany, 1976.54. Glass, L.; Mackey, M.C. From Clocks to Chaos. The Rhythms of Life; Princeton University Press: Princeton, NJ, USA, 1988.55. Schuster, H.G. Deterministic Chaos: An Introduction; VCH: Weinheim, Germany, 1988.56. Zhang, W.-B. Synergetic Economics; Springer: Berlin/Heidelberg, Germany, 1991.57. Anishchenko, V.S. Dynamical Chaos—Models and Experiments; World Scientific: Singapore, 1995.58. Abarbanel, H.D.I.; Rabinovich, M.I.; Selverston, A.; Bazhenov, M.V.; Huerta, R.; Sushchik, M.M.; Rubchinskii, L.L. Synchronisation

in neural networks. Phys. Usp. 1996, 39, 337. [CrossRef]59. Guckenheimer, J.; Hoffman, K.; Weckesser, W. Numerical computation of canards. Int. J. Bifurc. Chaos 2000, 10, 2669–2687.

[CrossRef]60. Rowat, P.F.; Selverston, A.I. Modeling the gastric mill central pattern generator of the lobster with a relaxation-oscillator network.

J. Neurophysiol. 1993, 70, 1030–1053. [CrossRef] [PubMed]61. Cartwright, J.; Eguiluz, V.; Hernandez-Garcia, E.; Piro, O. Dynamics of elastic excitable media. Int. J. Bifurc. Chaos 1999, 9,

2197–2202. [CrossRef]62. Slavin, A.; Tiberkevich, V. Nonlinear Auto-Oscillator Theory of Microwave Generation by Spin-Polarized Current. IEEE Trans.

Magn. 2009, 45, 1875–1918. [CrossRef]63. Baladron, J.; Fasoli, D.; Faugeras, O.; Touboul, J. Mean-field description and propagation of chaos in networks of Hodgkin-Huxley

and FitzHugh-Nagumo neurons. J. Math. Neurosci. 2012, 2, 10. [CrossRef]64. Braga, M.H.; Murchison, A.J.; Ferreira, J.A.; Singh, P.; Goodenough, J.B. Glass-amorphous alkali-ion solid electrolytes and their

performance in symmetrical cells. Energy Environ. Sci. 2016, 9, 948–954. [CrossRef]65. Braga, M.H.; Ferreira, J.A.; Murchison, A.J.; Goodenough, J.B. Electric Dipoles and Ionic Conductivity in a Na+Glass Electrolyte. J.

Electrochem. Soc. 2017, 164, A1–A7. [CrossRef]

Materials 2021, 14, 2398 16 of 16

66. Braga, M.H.; Murchison, A.J.; Oliveira, J.E.; Goodenough, J.B. Low-Temperature Performance of a Ferroelectric Glass ElectrolyteRechargeable Cell. ACS Appl. Energy Mater. 2019, 2, 4943–4953. [CrossRef]

67. Braga, M.H.; Subramaniyam, C.M.; Murchison, A.J.; Goodenough, J.B. Nontraditional, Safe, High Voltage Rechargeable Cells ofLong Cycle Life. J. Am. Chem. Soc. 2018, 140, 6343–6352. [CrossRef]

68. Braga, M.H.; Murchison, A.J.; Goodenough, J.B. Dataset on a ferroelectric based electrostatic and electrochemical Li-cell with atraditional cathode. Data Brief 2020, 29, 105087. [PubMed]

69. Braga, M.H.; Oliveira, J.E.; Murchison, A.J.; Goodenough, J.B. Performance of a ferroelectric glass electrolyte in a self-chargingelectrochemical cell with negative capacitance and resistance. Appl. Phys. Rev. 2020, 7, 011406. [CrossRef]

70. Videos and Photos of Experiments. Available online: https://paginas.fe.up.pt/~mbraga/ (accessed on 18 April 2021).71. Braga, M.H.; Oliveira, J.E. Ferroelectric Energy Harvest and Storage Cell; Mendeley Data. 2021; Volume 1, (video). [CrossRef]72. Braga, M.H.; Oliveira, J.E. Ferroelectric Energy Harvest and Storage Cell—II.; Mendeley Data. 2021; Volume 1, (video). [CrossRef]73. Braga, M.H.; Danzi, F. Ferroelectric Energy Harvest and Storage Cell—III.; Mendeley Data. 2021; Volume 1, (video). [CrossRef]74. Faber, T.E. An Introduction to the Theory of Liquid Metals; Cambridge University Press: Cambridge, UK, 1972.75. Kresse, G.; Furthmuller, J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev.

B 1996, 54, 11169. [CrossRef] [PubMed]76. Blochl, P.E. Projector augmented-wave method. Phys. Rev. B 1994, 50, 17953. [CrossRef] [PubMed]77. Perdew, J.P.; Wang, Y. Accurate and simple analytic representation of the electron-gas correlation energy. Phys. Rev. B 1992,

45, 13244.78. Nosé, S. Constant Temperature Molecular Dynamics Methods. Prog. Theor. Phys. Suppl. 1991, 103, 1–46. [CrossRef]79. Braga, M.H.; Ferreira, J.A.; Stockhausen, V.; Oliveira, J.E.; ElAzab, A. Novel Li3ClO based glasses with superionic properties for

lithium batteries. J. Mater. Chem. A 2014, 2, 5470–5480.